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Common fixed point theorems for expansion mappings in various abstract spaces using the concept of weak reciprocal continuity
Fixed Point Theory and Applications volume 2012, Article number: 221 (2012)
Abstract
In this paper, we prove expansion mapping theorems using the concept of compatible maps, weakly reciprocal continuity, Rweakly commuting mappings, Rweakly commuting of type ({A}_{f}), ({A}_{g}) and (P) in metric spaces and in Gmetric spaces.
MSC:54H25, 47H10.
1 Metric spaces
In 1922, Banach proved a common fixed point theorem which ensures, under appropriate conditions, the existence and uniqueness of a fixed point. This result of Banach is known as Banach’s fixed point theorem or Banach contraction principle. Many authors have extended, generalized and improved Banach’s fixed point theorem in different ways.
Jungck [1] proved a common fixed point theorem for commuting maps, which generalized the Banach fixed point theorem. This theorem has had many applications but suffers from one drawback that the continuity of a map throughout the space is needed. Jungck [2] defined the concept of compatible mappings.
Definition 1 ([2], see also [3])
A pair of selfmappings (f,g) of a metric space (X,d) is said to be compatible if {lim}_{n\to \mathrm{\infty}}d(fg{x}_{n},gf{x}_{n})=0 whenever \{{x}_{n}\} is a sequence in X such that {lim}_{n\to \mathrm{\infty}}f({x}_{n})={lim}_{n\to \mathrm{\infty}}g({x}_{n})=z for some z in X.
In 1994, Pant [4] introduced the notion of pointwise Rweak commutativity in metric spaces.
Definition 2 ([4], see also [5])
A pair of selfmappings (f,g) of a metric space (X,d) is said to be Rweakly commuting at a point x in X if d(fgx,gfx)\le Rd(fx,gx) for some R>0.
Definition 3 ([4])
Two selfmaps f and g of a metric space (X,d) are called pointwise Rweakly commuting on X if, given x in X, there exists R>0 such that d(fgx,gfx)\le Rd(fx,gx).
In 1997, Pathak et al. [6] generalized the notion of Rweakly commuting mappings to Rweakly commuting mappings of type ({A}_{f}) and of type ({A}_{g}).
Definition 4 ([6])
Two selfmaps f and g of a metric space (X,d) are called Rweakly commuting of type ({A}_{g}) if there exists some R>0 such that d(ffx,gfx)\le Rd(fx,gx) for all x in X.
Similarly, two selfmappings f and g of a metric space (X,d) are called Rweakly commuting of type ({A}_{f}) if there exists some R>0 such that d(fgx,ggx)\le Rd(fx,gx) for all x in X.
It is obvious that pointwise Rweakly commuting maps commute at their coincidence points and pointwise Rweak commutativity is equivalent to commutativity at coincidence points. It may be noted that both compatible and noncompatible mappings can be Rweakly commuting of type ({A}_{g}) or ({A}_{f}) but converse may not be true.
Definition 5 ([6])
Two selfmaps f and g of a metric space (X,d) are called Rweakly commuting of type (P) if there exists some R>0 such that d(ffx,ggx)\le Rd(fx,gx) for all x in X.
In 1999, Pant [7] introduced a new continuity condition, known as reciprocal continuity, and obtained a common fixed point theorem by using the compatibility in metric spaces. He also showed that in the setting of common fixed point theorems for compatible mappings satisfying contraction conditions, the notion of reciprocal continuity is weaker than the continuity of one of the mappings. The notion of pointwise Rweakly commuting mappings increased the scope of the study of common fixed point theorems from the class of compatible to the wider class of pointwise Rweakly commuting mappings. Subsequently, several common fixed point theorems have been proved by combining the ideas of Rweakly commuting mappings and reciprocal continuity of mappings in different settings.
Definition 6 ([7])
Two selfmappings f and g are called reciprocally continuous if {lim}_{n\to \mathrm{\infty}}fg{x}_{n}=fz and {lim}_{n\to \mathrm{\infty}}gf{x}_{n}=gz, whenever \{{x}_{n}\} is a sequence such that {lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=z for some z in X.
If f and g are both continuous, then they are obviously reciprocally continuous, but the converse is not true.
Recently, Pant et al. [8] generalized the notion of reciprocal continuity to weak reciprocal continuity as follows.
Definition 7 ([8])
Two selfmappings f and g are called weakly reciprocally continuous if {lim}_{n\to \mathrm{\infty}}fg{x}_{n}=fz or {lim}_{n\to \mathrm{\infty}}gf{x}_{n}=gz whenever \{{x}_{n}\} is a sequence such that {lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=z for some z in X.
If f and g are reciprocally continuous, then they are obviously weak reciprocally continuous, but the converse is not true. Now, as an application of weak reciprocal continuity, we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings which also includes noncompatible mappings.
Theorem 1 Let f and g be two weakly reciprocally continuous selfmappings of a complete metric space (X,d) satisfying the following conditions:
for any x,y\in X and q>1, we have that
If f and g are either compatible or Rweakly commuting of type ({A}_{g}) or Rweakly commuting of type ({A}_{f}) or Rweakly commuting of type (P), then f and g have a unique common fixed point.
Proof Let {x}_{0} be any point in X. Since g(X)\subseteq f(X), there exists a sequence of points \{{x}_{n}\} such that g({x}_{n})=f({x}_{n+1}).
Define a sequence \{{y}_{n}\} in X by
Now, we will show that \{{y}_{n}\} is a Cauchy sequence in X. For proving this, from (1.2), we have
Hence,
Therefore, for all n,m\in \mathbb{N} (a set of natural numbers), n<m, we have
Thus, \{{y}_{n}\} is a Cauchy sequence in X. Since X is complete, there exists a point z in X such that {lim}_{n\to \mathrm{\infty}}{y}_{n}=z. Therefore, by (1.3), we have {lim}_{n\to \mathrm{\infty}}{y}_{n}={lim}_{n\to \mathrm{\infty}}g({x}_{n})={lim}_{n\to \mathrm{\infty}}f({x}_{n+1})=z.
Suppose that f and g are compatible mappings. Now, by the weak reciprocal continuity of f and g, we obtain {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz, then the compatibility of f and g gives {lim}_{n\to \mathrm{\infty}}d(fg{x}_{n},gf{x}_{n})=0, that is, {lim}_{n\to \mathrm{\infty}}d(gf{x}_{n},fz)=0.
Hence, {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=fz. From (1.3), we get {lim}_{n\to \mathrm{\infty}}gf({x}_{n+1})={lim}_{n\to \mathrm{\infty}}gg({x}_{n})=fz.
Therefore, from (1.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, fz=gz. Again, the compatibility of f and g implies the commutativity at a coincidence point. Hence, gfz=fgz=ffz=ggz. Using (1.2), we obtain
which proves that gz=ggz. We also get gz=ggz=fgz and then gz is a common fixed point of f and g.
Next, suppose that {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. The assumption g(X)\subseteq f(X) implies that gz=fu for some u\in X and therefore, {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=fu.
The compatibility of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fu. By virtue of (1.3), we have {lim}_{n\to \mathrm{\infty}}gf({x}_{n+1})={lim}_{n\to \mathrm{\infty}}gg({x}_{n})=fu. Using (1.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Then we get fu=gu. The compatibility of f and g yields fgu=ggu=ffu=gfu. Finally, using (1.2), we obtain
that is, gu=ggu. We also have gu=ggu=fgu and gu is a common fixed point of f and g.
Now, suppose that f and g are Rweakly commuting of type ({A}_{f}). Now, the weak reciprocal continuity of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let us first assume that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz. Then the Rweak commutativity of type ({A}_{f}) of f and g yields
and therefore
This proves that {lim}_{n\to \mathrm{\infty}}gg{x}_{n}=fz. Again, using (1.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, we get fz=gz. Again, by using the Rweak commutativity of type ({A}_{f}), we have
This yields ggz=fgz. Therefore, ffz=fgz=gfz=ggz. Using (1.2), we get
that is, gz=ggz. Then we also get gz=ggz=fgz and gz is a common fixed point of f and g.
Similar proof works in the case where {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz.
Suppose that f and g are Rweakly commuting of type ({A}_{g}). Again, as done above, we can easily prove that fz is a common fixed point of f and g.
Finally, suppose that f and g are Rweakly commuting of type (P). The weak reciprocal continuity of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let us assume that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz. Then the Rweak commutativity of type (P) of f and g yields
Taking the limit as n\to \mathrm{\infty}, we get
that is, {lim}_{n\to \mathrm{\infty}}d(ff{x}_{n},gg{x}_{n})=0.
Using (1.1) and (1.3), we have, fg{x}_{n1}=ff{x}_{n}\to fz as n\to \mathrm{\infty}, which gives gg{x}_{n}\to fz as n\to \mathrm{\infty}. Also, using (1.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, fz=gz. Again, by using the Rweak commutativity of type (P),
This yields ffz=ggz.
Therefore, ffz=fgz=gfz=ggz. Using (1.2), we get
This proves that gz=ggz. Hence, gz=ggz=fgz and gz is a common fixed point of f and g.
Similar proof works in the case where {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz.
Uniqueness of the common fixed point theorem follows easily in each of the four cases by using (1.2). □
2 Gmetric spaces
In 1963, Gahler [9] introduced the concept of 2metric spaces and claimed that a 2metric is a generalization of the usual notion of a metric, but some authors proved that there is no relation between these two functions. It is clear that in 2metric, d(x,y,z) is to be taken as the area of the triangle with vertices x, y and z in {R}^{2}. However, Hsiao [10] showed that for every contractive definition, with {x}_{n}={T}^{n}{x}_{0}, every orbit is linearly dependent, thus rendering fixed point theorems in such spaces trivial.
In 1992, Dhage [11] introduced the concept of a Dmetric space. The situation for a Dmetric space is quite different from that for 2metric spaces. Geometrically, a Dmetric D(x,y,z) represents the perimeter of the triangle with vertices x, y and z in {R}^{2}. Recently, Mustafa and Sims [10] have shown that most of the results concerning Dhage’s Dmetric spaces are invalid. Therefore, they introduced an improved version of the generalized metric space structure, which they called Gmetric spaces.
In 2006, Mustafa and Sims [12] introduced the concept of Gmetric spaces as follows.
Definition 8 ([12])
Let X be a nonempty set, and let G:X\times X\times X\to {R}^{+} be a function satisfying the following axioms:
(G1) G(x,y,z)=0 if x=y=z,
(G2) 0<G(x,x,y) for all x,y\in X with x\ne y,
(G3) G(x,x,y)\le G(x,y,z) for all x,y,z\in X with z\ne y,
(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots (symmetry in all three variables),
(G5) G(x,y,z)\le G(x,a,a)+G(a,y,z) for all x,y,z,a\in X (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a Gmetric on X and the pair (X,G) is called a Gmetric space.
Definition 9 ([12])
Let (X,G) be a Gmetric space and let \{{x}_{n}\} be a sequence of points in X. A point x in X is said to be the limit of the sequence \{{x}_{n}\}, {lim}_{m,n\to \mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0, and one says that the sequence \{{x}_{n}\} is Gconvergent to x.
Thus, {x}_{n}\to x, n\to \mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}{x}_{n}=x in a Gmetric space (X,G) if for each \epsilon >0, there exists a positive integer N such that G(x,{x}_{n},{x}_{m})<\epsilon for all m,n\ge N.
Now, we state some results from the papers [10, 12–15] which are helpful for proving our main results.
Proposition 1 ([12])
Let (X,G) be a Gmetric space. Then the following are equivalent:

(1)
\{{x}_{n}\} is G convergent to x,

(2)
G({x}_{n},{x}_{n},x)\to 0 as n\to \mathrm{\infty},

(3)
G({x}_{n},x,x)\to 0 as n\to \mathrm{\infty},

(4)
G({x}_{m},{x}_{n},x)\to 0 as m,n\to \mathrm{\infty}.
Definition 10 ([12])
Let (X,G) be a Gmetric space. A sequence \{{x}_{n}\} is called GCauchy if, for each \epsilon >0, there exists a positive integer N such that G({x}_{n},{x}_{m},{x}_{l})<\epsilon for all n,m,l\ge N; i.e., if G({x}_{n},{x}_{m},{x}_{l})\to 0 as n,m,l\to \mathrm{\infty}.
Proposition 2 ([15])
If (X,G) is a Gmetric space, then the following are equivalent:

(1)
the sequence \{{x}_{n}\} is GCauchy,

(2)
for each \epsilon >0, there exists a positive integer N such that G({x}_{n},{x}_{m},{x}_{m})<\epsilon for all n,m\ge N.
Proposition 3 ([12])
Let (X,G) be a Gmetric space. Then the function G(x,y,z) is jointly continuous in all three of its variables.
Definition 11 ([12])
A Gmetric space (X,G) is called a symmetric Gmetric space if
Proposition 4 ([14])
Every Gmetric space (X,G) will define a metric space (X,{d}_{G}) by

(i)
{d}_{G}(x,y)=G(x,y,y)+G(y,x,x) for all x, y in X.
If (X,G) is a symmetric Gmetric space, then

(ii)
{d}_{G}(x,y)=2G(x,y,y) for all x, y in X.
However, if (X,G) is not symmetric, then it follows from the Gmetric properties that

(iii)
\frac{3}{2}G(x,y,y)\le {d}_{G}(x,y)\le 3G(x,y,y) for all x, y in X.
Definition 12 ([14])
A Gmetric space (X,G) is said to be Gcomplete if every GCauchy sequence in (X,G) is Gconvergent in X.
Proposition 5 ([14])
A Gmetric space (X,G) is Gcomplete if and only if (X,{d}_{G}) is a complete metric space.
Proposition 6 ([15])
Let (X,G) be a Gmetric space. Then, for any x,y,z,a\in X, it follows that

(i)
if G(x,y,z)=0, then x=y=z,

(ii)
G(x,y,z)\le G(x,x,y)+G(x,x,z),

(iii)
G(x,y,y)\le 2G(y,x,x),

(iv)
G(x,y,z)\le G(x,a,z)+G(a,y,z),

(v)
G(x,y,z)\le \frac{2}{3}(G(x,y,a)+G(x,a,z)+G(a,y,z)),

(vi)
G(x,y,z)\le (G(x,a,a)+G(y,a,a)+G(z,a,a)).
Definition 13 ([16])
A pair of selfmappings (f,g) of a Gmetric space (X,G) is said to be compatible if {lim}_{n\to \mathrm{\infty}}G(fg{x}_{n},gf{x}_{n},gf{x}_{n})=0 or {lim}_{n\to \mathrm{\infty}}G(gf{x}_{n},fg{x}_{n},fg{x}_{n})=0 whenever \{{x}_{n}\} is a sequence in X such that {lim}_{n\to \mathrm{\infty}}f({x}_{n})={lim}_{n\to \mathrm{\infty}}g({x}_{n})=z for some z in X.
Definition 14 ([17])
A pair of selfmappings (f,g) of a Gmetric space (X,G) is said to be Rweakly commuting at a point x in X if G(fgx,gfx,gfx)\le RG(fx,gx,gx) for some R>0.
Definition 15 ([17])
Two selfmaps f and g of a Gmetric space (X,G) are called pointwise Rweakly commuting on X if, given x in X, there exists R>0 such that G(fgx,gfx,gfx)\le RG(fx,gx,gx).
Definition 16 ([6])
Two selfmaps f and g of a Gmetric space (X,G) are called Rweakly commuting of type ({A}_{g}) if there exists some R>0 such that G(ffx,gfx,gfx)\le RG(fx,gx,gx) for all x in X. Similarly, two selfmappings f and g of a Gmetric space (X,G) are called Rweakly commuting of type ({A}_{f}) if there exists some R>0 such that G(fgx,ggx,ggx)\le RG(fx,gx,gx) for all x in X.
Definition 17 ([6])
Two selfmappings f and g of a Gmetric space (X,G) are called Rweakly commuting of type (P) if there exists some R>0 such that G(ffx,ggx,ggx)\le RG(fx,gx,gx) for all x in X.
Theorem 2 Let f and g be two weakly reciprocally continuous selfmappings of a complete Gmetric space (X,G) satisfying the following conditions:
for any x,y,z\in X and q>1, we have that
If f and g are either compatible or Rweakly commuting of type ({A}_{g}) or Rweakly commuting of type ({A}_{f}) or Rweakly commuting of type (P), then f and g have a unique common fixed point.
Proof Let {x}_{0} be any point in X. Since g(X)\subseteq f(X), there exists a sequence of points \{{x}_{n}\} such that g({x}_{n})=f({x}_{n+1}).
Define a sequence \{{y}_{n}\} in X by
Now, we will show that \{{y}_{n}\} is a GCauchy sequence in X. For proving this, by (2.2) take x={x}_{n}, y={x}_{n+1}, z={x}_{n+1}, we have
Continuing in the same way, we have
Therefore, for all n,m\in \mathbb{N} (a set of natural numbers), n<m, we have by using (G5)
Thus, \{{y}_{n}\} is a GCauchy sequence in X. Since (X,G) is a complete Gmetric space, there exists a point z in X such that {lim}_{n\to \mathrm{\infty}}{y}_{n}=z. Therefore, by (2.3), we have {lim}_{n\to \mathrm{\infty}}{y}_{n}={lim}_{n\to \mathrm{\infty}}g({x}_{n})={lim}_{n\to \mathrm{\infty}}f({x}_{n+1})=z.
Suppose that f and g are compatible mappings. Now, the weak reciprocal continuity of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz, then the compatibility of f and g gives {lim}_{n\to \mathrm{\infty}}G(gf{x}_{n},fg{x}_{n},fg{x}_{n})=0, that is, G({lim}_{n\to \mathrm{\infty}}gf{x}_{n},fz,fz)=0.
Hence, {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=fz. From (2.3), we get {lim}_{n\to \mathrm{\infty}}gf({x}_{n+1})={lim}_{n\to \mathrm{\infty}}gg({x}_{n})=fz.
Therefore, by (2.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, fz=gz. Again, the compatibility of f and g implies the commutativity at a coincidence point. Hence, gfz=fgz=ffz=ggz. Now, we claim that gz=ggz. Suppose not, then by using (2.2), we obtain
and
which gives contradiction because q>1. Hence, gz=ggz. Hence, gz=ggz=fgz and gz is a common fixed point of f and g.
Next suppose that {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. The assumption g(X)\subseteq f(X) implies that gz=fu for some u\in X and therefore, {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=fu.
The compatibility of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fu. By virtue of (2.3), this gives {lim}_{n\to \mathrm{\infty}}gf({x}_{n+1})={lim}_{n\to \mathrm{\infty}}gg({x}_{n})=fu. Using (2.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
which gives fu=gu. The compatibility of f and g yields fgu=ggu=ffu=gfu. Finally, we claim that gu=ggu. Suppose not, then by using (2.2), we obtain
and
that is, gu=ggu. Hence, gu=ggu=fgu and gu is a common fixed point of f and g.
Now, suppose that f and g are Rweakly commuting of type ({A}_{f}). Now, the weak reciprocal continuity of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let us first assume that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz. Then the Rweak commutativity of type ({A}_{f}) of f and g yields
Taking the limit as n\to \mathrm{\infty}, we get
This proves that {lim}_{n\to \mathrm{\infty}}gg{x}_{n}=fz. Again, using (2.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, we get fz=gz. Again, by using the Rweak commutativity of type ({A}_{f}),
This yields ggz=fgz. Therefore, ffz=fgz=gfz=ggz. We claim that gz=ggz. Suppose not, using (2.2), we get
and
a contradiction, that is, gz=ggz. Hence, gz=ggz=fgz and gz is a common fixed point of f and g.
Similar proof works in the case where {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz.
Suppose that f and g are Rweakly commuting of type ({A}_{g}). Again, as done above, we can easily prove that gz is a common fixed point of f and g.
Finally, suppose f and g are Rweakly commuting of type (P). The weak reciprocal continuity of f and g implies that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz or {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz. Let us assume that {lim}_{n\to \mathrm{\infty}}fg({x}_{n})=fz. Then the Rweak commutativity of type (P) of f and g yields
Taking the limit as n\to \mathrm{\infty}, we get
That is,
Using (2.1) and (2.3), we have fg{x}_{n1}=ff{x}_{n}\to fz as n\to \mathrm{\infty}, which gives gg{x}_{n}\to fz as n\to \mathrm{\infty}. Also, using (2.2), we get
Taking the limit as n\to \mathrm{\infty}, we get
Hence, fz=gz. Again, by using the Rweak commutativity of type (P),
This yields ffz=ggz.
Therefore, ffz=fgz=gfz=ggz. Finally, we claim that gz=ggz. Suppose not, using (2.2), we get
and
a contradiction, we get gz=ggz. Hence, gz=ggz=fgz and gz is a common fixed point of f and g.
Similar proof works in the case where {lim}_{n\to \mathrm{\infty}}gf({x}_{n})=gz.
Uniqueness of the common fixed point theorem follows easily in each of the four cases by using (2.2). □
We now give an example (see also [18]) to illustrate Theorem 2.
Example 1 Let (X,G) be a Gmetric space, where X=[2,20] and
for all x,y,z\in X. Define f,g:X\to X by
Let \{{x}_{n}\} be a sequence in X such that either {x}_{n}=2 or {x}_{n}=5+1/n for each n.
Then, clearly, f and g satisfy all the conditions of Theorem 2 and have a unique common fixed point at x=2.
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Acknowledgements
The authors thank the editor and the referees for their useful comments and suggestions. The second author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand’s Office of the Higher Education Commission for financial support.
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Manro, S., Kumam, P. Common fixed point theorems for expansion mappings in various abstract spaces using the concept of weak reciprocal continuity. Fixed Point Theory Appl 2012, 221 (2012). https://doi.org/10.1186/168718122012221
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DOI: https://doi.org/10.1186/168718122012221
Keywords
 compatible maps
 metric spaces
 Gmetric spaces
 weakly reciprocal continuity
 Rweakly commuting mappings
 Rweakly commuting of type ({A}_{f}), ({A}_{g}) and (P)